In a blender, make the pesto by combining the avocado, basil leaves, Parmesan cheese, garlic, pistachios, ½ teaspoon salt and ¼ teaspoon pepper.
Pulse until the mixture is roughly chopped, then with the blender running, slowly stream in the olive oil.
Continue blending the pesto until it is a thick paste, scraping down the sides of the blender as needed. Set the pesto aside.
Preheat the grill for 10 minutes on medium heat. Spread 1 tablespoon butter on each ear of corn then wrap them in aluminum Foil.
Place the corn on the grill, close the lid and grill the corn, turning it frequently, until it is tender, 15 to 20 minutes. Unwrap the corn, and once it is cool enough to handle, slice the kernels off the cob.
Brush both sides of the flatbreads with olive oil then grill them for 1 to 2 minutes on each side.
Remove the flatbreads from the grill and top them with the avocado pesto, leaving a 1/2-inch border around the edges.
Top the pesto with the Gruyere cheese, mozzarella cheese, red onions and corn. Return the flatbreads to the grill, close the lid and grill them until the cheese is melted.
Remove the flatbreads from the grill and transfer them to serving plates. Top the flatbreads with the arugula